Kendall random walks

TL;DR

This paper introduces a new class of Markov random walks based on Kendall convolution, exploring their properties, fluctuations, and connection to Pareto distributions, and presents a novel technique involving a modified Williamson transform.

Contribution

It develops a new framework for random walks using Kendall convolution, detailing their properties, fluctuations, and relation to Pareto-type heavy-tailed distributions, which are not classical Lévy processes.

Findings

Kendall convolution generates new heavy-tailed Pareto-type distributions.

Random walks with Kendall convolution are not classical Lévy processes.

A new technique using a modified Williamson transform is proposed for distribution curation.

Abstract

The paper deals with a new class of random walks strictly connected with the Pareto distribution. We consider stochastic processes in the sense of generalized convolution or weak generalized convolution following the idea given in [1]. The processes are Markov processes in the usual sense. Their structure is similar to perpetuity or autoregressive model. We prove theorem, which describes the magnitude of the fluctuations of random walks generated by generalized convolutions. We give a construction and basic properties of random walks with respect to the Kendall convolution. We show that they are not classical L\'evy processes. The paper proposes a new technique to cumulate the Pareto-type distributions using a modification of the Williamson transform and contains many new properties of weakly stable probability measure connected with the Kendall convolution. It seems that the Kendall convolution produces a new classes of heavy tailed distributions of Pareto-type.